There's no other potential meaning. The US Senate is under Democratic majority leadership, and Republican special interest group NOM wishes it would be in the hands of conservatives instead. That's because NOM is actually deeply partisan, as are all of its key staffers.

When they talk about being bi- or nonpartisan, all they mean is that they will take support, money, votes, etc. from Democrats. But let's be clear about this: those Democrats who do sign on to the NOM agenda do so at the expense of their own party and its interests.